Tag: geometry

Double Square and the Golden Rectangle

above: Dan Palmateer wrote of this, “it just hit me that the conjunction of the circle to the golden rectangle existed.”

Here we will continue in the mode of a lesson in Geometry where what is grasped intuitively has to have reason for it to be true. It occurred to me that the square in the top hemisphere is the twin of a square in the lower hemisphere, hence this has a relationship to the double square rectangle. So one can (1) Make a Double Square and then (2) Find the center and (3) a radius can then draw the out-circle of a double square (see diagram below).

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Geometry 7: Geometrical Expansion

above: the dolmen of Pentre Ifan (wiki tab)

In previous lessons, fixed lengths have been divided into any number of equal parts, to serve the notion of integer fractions in which the same length can then be reinterpreted as to its units or as a numerically different measurement. This allows all sorts of rescaling and exploitation of the properties of integer numbers.

Here we present a megalithic method which extended two or more fixed bearings (or alignments), usually based upon a simple geometrical form such as a triangle or a rectangle. This can be how the larger geometries came to be drawn on the landscape (here called landforms) of separated megaliths and natural features which appear to belong together. For example,

Outliers: Alexander Thom found that British stone circle were often associated with single outliers (standing stones) on a bearing that may correspond to horizon event but equally, appears to give clues to the metrology of the circle in the itinerary length to the outlier from the circle’s centre.

Figure 1 Stone circle plans often indicate nearby outliers and stone circles

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Geometry 6: the Geometrical AMY

By 2016 it was already obvious that the lunar month (in days) and the PMY, AMY and yard (in inches) had peculiar relationships involving the ratio 32/29, shown above. This can now be explained as a manifestation of day-inch counting and the unusual numerical properties of the solar and lunar year, when seen using day-inch counting.

It is hard to imagine that the English foot arose from any other process than day-inch counting; to resolve the excess of the solar year over the lunar year, in three years – the near-anniversary of sun and moon. This created the Proto Megalithic Yard (PMY) of 32.625 day-inches as the difference.

Figure 1 The three solar year count’s geometrical demonstration of the excess in length of 3 solar years over 3 lunar years as the 32.625 day-inch PMY.

A strange property of N:N+1 right triangles can then transform this PMY into the English foot, when counting over a single lunar and solar year using the PMY to count months.

The metrological explanation

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Le Site Mégalithique du Manio à Carnac

by Howard Crowhurst

Perched on a hill in the forest north of the Carnac alignments, a megalithic site has escaped the fences that have littered the landscapes of the region for several years. These are the menhir and the quadrilateral of Manio. From the outset, the large menhir impresses with its dimensions. Nearly 5m50 high, it is the highest standing stone in the town.

More discreet, the quadrilateral caps the top. 90 upright and contiguous stones, varying in height between 10 cm and 1m60, make up an enclosure approximately 36 meters long and 8 meters wide on average, because the long sides converge. The stones at the ends draw a curve. Four stones to the northeast form the remains of a circle. Two menhirs, much larger than all the other stones in the quadrilateral, open a kind of door in the south file. This particular form questions us. What could she be used for? Was it a meeting place, maybe an enclosure for sheep? In fact, what we see today is probably only the outer skeleton of a larger monument, a mound of stone and earth that contained a chamber inside. Other remains complicate the whole, unless they help us solve our puzzle. Hidden in the brambles and brush, we can discover a stone on the ground of rounded shape. These curves are reminiscent of the belly of a pregnant woman. She is nicknamed the “Lady” of the Manio.

Use of foot ratios in Megalithic Astronomy

The ratios of ancient metrology emerged from the Megalithic innovations of count&compare: counting time as length and comparing lengths as the longest sides of right triangles. To compare two lengths in this way, one can take a longer rope length and lay it out (say East-West), starting at the beginning of the shorter rope length, using a stake in the ground to fix those ends together.

The longer rope end is then moved to form an angle to the shorter, on the ground, whilst keeping the longer rope straight. The Right triangle will be formed when the longer rope’s end points exactly to the North of the shorter rope end. But to do that one needs to be able to form a right angle at the shorter rope’s end. The classic proposal (from Robin Heath) is to form the simplest Pythagorean triangle with sides {3 4 5} at the rope’s end. One tool for this could then have been the romantic knotted belt of a Druid, whose 13 equally spaced knots could define 12 equal intervals. Holding the 5th knot, 8th knot and the starting and ending knots together automatically generates that triangle sides{3 4 5}.

Forming a square with the AMY is helped by the diagonals being rational at 140/99 of the AMY.
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Old Yard’s Mastery of the Square Root of 2

The old yard was almost identical to the yard of three feet, but just one hundredth part smaller at 2.87 feet. This gives its foot value as 99/100 feet, a value belonging to a module very close to the English/Greek which defines one relative to the rational ratios of the Historical modules.

So why was this foot and its yard important, in the Scottish megalithic and in later, historical monuments?

If one forms a square with side equal to the old yard, that square can be seen as containing 9 square feet, and each of those has side length 99/100 feet. This can be multiplied by the rough approximation to 1/√ 2 of 5/7 = 0.714285, to obtain a more accurate 1/√ 2 of 99/140 = 0.70714285.

Figure 1 Forming a Square with the Old Yard. The diagonal of the foot squares is then 7/5, the simplest approximation to √ 2.
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